Acta mater. Vol. 45, No. 11, pp. 4695-4702, 1997
0 1997 Acta Metallurgica Inc.
Pergamon
Published by ElsevierScience Ltd. All rights reserved
Printed in Great Britain
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TOUGHNESS VARIATION OF FERROELECTRICS
BY
POLARIZATION
SWITCH UNDER NON-UNIFORM
ELECTRIC FIELD
TING ZHU and WE1 YANG?
Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
(Received 23 October
1996; accepted
11 March
1997)
Abstract-Low
fracture toughness of ferroelectric ceramics calls for a reliability concern in the
development of smart structures. For ferroelectrics under mechanical and electrical loadings, the
intensified stress and electric fields in the vicinity of a crack-like flaw lead to domain reorientation. The
switched domains induce incompatible strain near the flaw and consequently change the apparent fracture
toughness. The present paper investigates toughness variation by small scale polarization switching under
non-uniform electric and stress fields. A two-term near tip electric field is obtained by analyzing a
permeable elliptical flaw in a ferroelectric solid. Analytical solution for the shielding stress intensity factor
is derived for the ideal situation of mono-domain ferroelectrics. The result is generated by a Reuss-type
assembly to the poly-domain ferroelectrics. The predictions based on the present model explain the
conflicting experimental data on the toughness variations with respect to the applied electric field. 0 1997
Acta Metallurgica
Inc.
1. INTRODUCTION
Advances in modern technology set the ferroelectric
ceramics at the center stage in the development
of
sensors and actuators in smart structures. However,
ferroelectric
ceramics are brittle and susceptible to
cracking, raising the need for investigation
of their
toughening mechanisms.
Experimental
and theoretical analyses [l-7] indicated that domain switching
plays an important role in the toughness variation of
ferroelectrics.
Under
mechanical
and electrical
loadings, the intensified stress and electric fields in the
vicinity
of a crack-like
flaw lead to domain
reorientation.
The switched domains induce incompatible strain under the constraint
of unswitched
material,
and consequently
alter the stress distribution near the flaw. The apparent
toughness
of
ferroelectrics thus varies due to domain switching. In
the experimental
studies, Mehta
and Virkar [2]
examined
domain switching by X-ray diffraction.
Lynch et al. [3] observed residual switching by use of
birefringence.
Theoretically,
switching
induced
toughness variation can be pursued in the same vein
as transformation
toughening
in ceramics,
see
McMeeking
and Evans [S], Budiansky
et al. [9].
Taking into account
the stress induced
domain
switching, Pisarenko et al. [l] and Mehta and Virkar
[2] studied fracture toughness
anisotropy
for ferroelectrics under pure mechanical loading. Recently,
Yang and Zhu [7] proposed a model to investigate
stress activated switching caused by uniform electric
~To whom all correspondence
should be addressed.
field for ferroelectrics
under combined mechanical
and electrical loadings.
Whether the electric field impedes or enhances
crack propagation
is still a debatable issue. Several
research groups [3, l&13] reported conflicting experimental
results.
Park and Sun [lo] performed
compact
tension
tests and found that apparent
fracture toughness varied asymmetrically
for poled
ferroelectrics
under positive and negative electric
fields. A positive field along the poling axis reduced
the fracture load, while a negative one increased it.
Tobin and Pak [ 1 l] and Lynch [4] reached the similar
conclusions
by using Vicker’s indentation.
On the
other hand, Singh and Wang [12] observed
an
opposite trend. Namely the crack propagates
less
under a positive applied field, and longer under a
negative
one. Park and Sun [lo] verified their
experimental
results
by a criterion
based
on
mechanical strain energy release rate. On the other
hand, considering
piezoelectric
effect, Kumar and
Singh [ 131 performed finite element analyses and their
results checked with the experimental
data of Singh
and Wang [ 121. To predict accurately the electric field
influence, the nonlinear effect of domain reorientation near the crack tip is important,
as pointed out
by Suo [5]. Considering
domain switching mechanism, Yang and Zhu [7] studied the toughening effect
under uniform electric field, and their prediction is
consistent with the observation of Park and Sun [lo].
Nevertheless,
the electric field in reality is non-uniform in the vicinity of the flaw. It is that non-uniform
electric field distribution
that may influence the
toughness variation.
4695
4696
ZHU and YANG:
TOUGHNESS
In this paper, a theoretical model for switching
induced toughness variation is presented, which
works for the situation of simultaneously applied
stress and electric fields. We consider the case that
both the stress and the electric fields are non-uniform.
The model explores in detail the effect of non-uniform electric field on the mechanical fracture process.
The plan of the paper is as follows. In the next section
the non-uniform fields outside the switching zone are
derived under the assumption of small scale domain
switching. We adopt the criterion of polarization
switch proposed by Hwang et al. [14] for an
individual domain, in which domains switch when the
combined (electric and mechanical) work reaches a
threshold value. In Section 3 we evaluate toughness
variation for mono-domain ferroelectrics. The analytical solution given there reveals several dimensionless material parameter groups of the problem. In
Section 4, we propose a Reuss type poly-domain
assembly to estimate the toughness variation of fully
poled ferroelectrics.
2. SMALL
SCALE
DOMAIN
SWITCHING
Attention is focused on the case of small scale
switching, in the sense that the specimen size is
considerably larger than the switching zone size. The
merit of the small scale switching model can be
appreciated as follows. Firstly, geometric complexities are separated from the domain switching process.
More importantly,
switching strain cannot be
relieved by global deformation in the case of small
scale switching, and thus contributes to toughen the
ceramics.
2.1. Non-uniform electric and stress fields
We now derive the non-uniform fields outside the
switching zone. As the first model to address this
issue, we try to simplify the problem while retaining
the key dimensionless material groups. Two assumptions are introduced: (1) the interaction between
stress and electric fields is weak outside the switching
zone; and (2) the material is modeled as elastically
and dielectrically isotropic. Therefore, the non-uniform fields outside the switching zone can be
determined by isotropic linear elasticity and electrostatics, respectively. The detailed structure of the
coupled nonlinear fields near the flaw tip, as well as
the slight anisotropy in material properties, are
neglected.
We first evaluate non-uniform electric field. One
controversial issue is the boundary conditions along
the crack surface. Impermeable condition has been
frequently adopted. However, results given by
McMeeking [15] on electrostrictive solids and by
Dunn 1161on piezoelectrics indicated that impermeable assumption may misinterpret the effects of the
electric field on crack propagation. In reality, the
small, but finite, crack volume contains air or some
other gas whose dielectric constant does not vanish.
OF FERROELECTRICS
Electric field may penetrate through the crack
volume. We derive the electric field based on the
solution for an elliptic flaw given by McMeeking [ 151.
As shown in Fig. 1, an electric field of strength E” is
imposed perpendicular to the flaw. The permittivities
of the elliptic flaw and the matrix are cf and cm,
respectively. Symbols a and b measure the lengths of
the semi-axes of the ellipse. For the case of vertically
applied remote electric field, the electric field outside
the flaw can be obtained from equation (9) in [15].
After some rearrangement, it becomes
1-K
i’ + l+K
-E,
where E, and E2 are the components
field in the x and y direction, and
z =
(1)
+ iE2 = IE”
of the electric
x + iy = i [(a + b)c + (a - b){-‘1,
(2)
and i = 0.
The dimensionless parameter K
represents the relative permeability of the flaw,
encompassing both the flaw geometry and the
dielectric parameters. The value of K is determined
by the relative magnitude of the permittivities ratio
tr/tm and the aspect ratio b/a. Consider the case of
b<<a but l/x is not necessarily small, equation (1)
reduces to
-j[z+pz]2+~
-El
+ iE2 = iE”
(4)
$z+Jq-1
where appropriate
branch has been selected. Taking
+E”
Fig. 1. Schematic
illustration
of an elliptical flaw in a
ferroelectric
solid subjected to applied electrical loading.
ZHU and YANG:
TOUGHNESS OF FERROELECTRICS
the Taylor expansion of the right hand side of (4) at
z = a and keeping the two leading terms, one obtains
-Et
+ iEz = iE”
Let the origin of the polar coordinate, r and 8, be
located at the right tip of the flaw, as shown in Fig.
1. The electric field outside the flaw is
(6)
The first term in (6) represents the singular field
governed
by
the
electric
intensity
factor
KE=EUr
m, and the second term the uniform
electric field. As can be seen from (6) the electric field
solution for an impermeable crack is recovered in the
vicinity of the flaw for the case of K << 1. If the aspect
ratio b/a is an order of magnitude larger than the
ratio of permittivities cf/t,, the impermeable crack
solution serves as a good approximation. The region
dominated by the first singular term shrinks as K
increases. At the other extreme of K >> 1, the uniform
electric field prevails in most regions around the crack
tip and the conducting crack solution is recovered. If
b/a is comparable to tr/t,,,, electric field effects of both
terms should be included.
The stress field near the switching zone boundary
may be approximated by the remote K field
3. MONO-DOMAIN SOLUTION
We start with the simple case of mono-domain
ferroelectrics. The solution obtained in this section
will serve as a fundamental
solution for the
poly-domain ferroelectrics. Consider the plane strain
situation. Suppose that all domains are oriented
along the poling direction after the polarization
process. Global coordinates X, and X, are introduced, such that X, directs along the longer axis of
the flaw and Xz normal to it. The uniform initial
poling direction forms an angle 4 with the X, axis.
3.1. Geometry of the switching zone
We first estimate the geometry of the switching
zone. Consider only 90” switching. There are two
possible variants of 90” switching: the domain can
switch 90” clockwise or anti-clockwise, with its
polarization vector P, rotating 90” clockwise or
anti-clockwise. In global coordinates, the polarization switch vector AP, can be expressed as
AP,=JP,
2.2. Switching criterion
An electric field may rotate the polar direction of
a domain by either 90 or 180 degrees, but a stress field
rotates it only by 90 degrees. A 180” switching does
not result in any strain, while a 90” switching results
in a fixed strain. Here we adopt the switching
criterion proposed by Hwang et al. [14] for an
individual domain. The criterion states that a domain
switches when the electrical work plus the mechanical
work exceed a critical value
u,At,, + EAR 2 2P&,
jco++‘-/.
L
/J
-cos 24 sin 24
sin 2q5 cos 24 I ’
(10)
where ys denotes the spontaneous strain associated
with 90” switching. Please note that the switching
strain is uniform in the entire switching zone, while
the stress and electric fields are not. Substituting the
stress field (7) the electric field (6) the polarization
switch vector (9) and the switching strain tensor (10)
into the switching criterion (8), one obtains
(8)
=2P,E,
where c,, is the stress tensor, At,, the switching strain
tensor, AP, the polarization switch vector, E, the
electric field vector, P, the spontaneous polarization
and EC the coercive field. The geometry of the
switching zone can be estimated by substituting the
stress and electric fields into (8).
\
(9)
The values of -$z and +$ in (9) correspond to 90”
switching clockwise and anti-clockwise, respectively.
The switching strain tensor AE, is the same for both
cases
AC,= ys
where Kdpp denotes the applied stress intensity factor
and o,(0) the universal stress angular distribution.
4697
1-E”K.sin
$E,lfK
>(11)
where r now outlines the boundary of the switched
domain from the crack tip. The shape of the
switching zone, in terms of a relation between
the radius r and the polar angle 6 can be expressed
ZHU and YANG:
4698
TOUGHNESS OF FERROELECTRICS
(a)
as follows
4
= &R(&
P, K),
(12)
0.16
0.12-
where the value
z
1 Kppys
‘=s?l
(3
P,E,
,.-----.__
:
(13)
gives a reference scale of the domain switching zone
around the crack tip, and the dimensionless
parameter
(14)
measures the relative magnitude
between the
non-uniform electric field and the non-uniform stress
field. The dimensionless function R(0; j?, K) in (12)
can be written explicitly as
0
-0.16
-0.24 -0.20 -0.16 -0.12 -0.08 -0.04 0.00
‘q/P
R(6; P, K) =
As mentioned above, -$r and +$t in (15)
correspond to clockwise and anti-clockwise 90”
switching. The actual switching selects the direction
with higher algebraic value of r. This can be
explained on the physical ground that switching
prefers the direction along which reorientation
requires less work. Figure 2(a) and (b) exhibit the
effects of electric fields on the geometry of the
switching zone. The two terms in the numerator of
(15) represent the influences from the singular electric
field and the stress field, respectively. The effect of the
uniform electric field is shown in the denominator of
(15). While the singular electric field changes both the
shape and the size of the switching zone, the uniform
electric field only modifies the size of the switching
zone. The relative contribution between the singular
electric field and the uniform electric field, namely the
first and the second terms in the expansion (6), is
measured by the parameter IC. For the case of
K = 103, uniform electric field dominates, and the
switching zone geometry is shown in Fig. 2(a). Since
the first term in the numerator of (15) is negligible,
the switching zone geometry becomes insensitive to
the values of /I. As the applied electric field varies, the
size of the switching zone varies, but not its shape. A
positive electric field reduces the size of the switching
zone, while a negative one enlarges it. For the
opposite case of K = 10m3, the effect of singular
electric field dominates, and the switching zone
geometry is shown in Fig. 2(b). Both the shape and
the size of the switching zone are changed by varying
the remote electric field, and either a positive or a
Fig. 2. Domain switching boundary of mono-domain
ferroelectrics [d = (s/2)] under positive and negative electric
fields. (a) K = 103;(b) K = 10d3.
negative electric field enlarges the size of switching
zone.
In formula (12), the positive nature of 4 requires
that R(B; /I, IC) is greater than zero. This condition
determines the initial angles, denoted as 0: and 8;)
for polarization switching above and below the crack
extension line. The terminating angles, 0: and f?;,
refer to the polar angles where the switching zone
above and below the crack reach the maximum
heights. These angles can be determined numerically.
3.2. Toughness variation for steady state crack growth
For dilatant transformation,
it was shown by
McMeeking and Evans [8] that Kappapproaches a
steady state value after a crack grows more than the
height of the transformation zone. In the transform
ZHU and YANG:
4699
TOUGHNESS OF FERROELECTRICS
where hi is the near-tip weight function,
ation toughening involving both dilatant and shear
contributions, rapid approaching to the steady state
asymptote is also anticipated, see Lambropoulos
[17]. In this work, we only evaluate the steady state
fracture toughness (Aa <<a), bearing in mind that it
serves as an upper bound.
We first derive the change in crack-tip stress
as a result of switch-induced
stress
intensity,
redistribution. Denote Kup as the intrinsic fracture
toughness that governs the fracture process near the
crack tip. For the ceramics in paraelectric phase,
there is no switch toughening. Hence, Kflpequals Kapp.
Since switching strain is deviatoric, one has
For the ceramics in ferroelectric phase, we relate Ktlp
to Kappby writing
Y
r,=Av, ,
1+v
Kup = Kapp+ AK.
(16)
The toughness increment AK can be evaluated
by an Eshelby technique, see [8]. The switching
strain induces a thin layer of accommodating
body forces T, on the boundary I, of the switching
zone. For a growing crack, the frontal switching
boundary translates as the crack advances, while the
wake remains immobile. AK is given by
AK =
T;h, dI,
(17)
(19)
where Y denotes the Young’s modulus and v the
Poisson’s ratio. The vector n, represents the outward
normal of I,. Substituting
(18), (19) and the
switching zone geometry (12) into (17) one obtains
the toughness variation,
where K&4)
corresponds to the applied stress
intensity factor to produce a near tip stress intensity
of Ktlp for a mono-domain ferroelectrics of orientation 4, and
2
r?=(l-3;,,,
(21)
(22)
sin0 sin 24 + 1! sin 24 + 28 (cos0 + 4v - 3)
(
2)
(
2 )
]:j
F(4)=[
-
(1 - v)[sin(44 + 38) - sin(44 + 0) - sin 281
i
+t sin(44 + 0) + & sin(44 + 20) - + sin(44 + 30)
-&
sin(4$ + 48) + k sin(44 + 60) i sin 8 + i sin 28
o:,e;
,
8:,s,-
(23)
sin (4 * ;i3n - 2“)sin (24 + z“)
(cos 0 + 4v - 3) ];
WP~=&{[
+os
(
+r+4e
0;8;
)I0:ll;
I
(24)
ZHU and YANG:
4700
TOUGHNESS
Table 1. Data used in numerical calculation [14]
P, =
0.30 C/m2
E = 0.36 MV/m
Y=SOGPa
Spontaneous polarization
Coercive field
Young’s modulus
Poisson’s ratio
Intrinsic fracture toughness
kz “‘0.6 MPaI&
The dimensionless group of material parameters q
emerges as the measure of the ratio between the
elastic strain energy and the threshold of switching
energy. The dimensionless functions CI($), D(4) and
F( 4) represent the influences of uniform electric field,
singular electric field, and stress field on the near tip
K field, respectively.
Substituting (20) into (16), one can write the
shielding ratio as follows
&v(d) _
D(6)
’- &g 2p$
KhP
(25)
l+ 87&)
F(4)
To evaluate the shielding ratio, we take a numerical
iterative scheme. Under a given electric field, an
initial Kapp(c$)/Ktt, is assumed. Then geometric
parameters 0: and 02 are determined numerically,
and those data are substituted into (23) and (24) to
evaluate F(4) and O(4). A new shielding ratio is
estimated according to (25). The procedure is
repeated until it converges. Material parameters used
in calculation are listed in Table 1. The length of
crack-like flaw 2a is assumed to be 2 mm and
spontaneous strain 0.003.
Figure 3 shows the variation of shielding ratio
as a function of the initial poling direction under
non-uniform
positive and negative electric field
(K = 1). A positive electric field directs along the
2.5 -
OF FERROELECTRICS
positive XZ axis, and a negative electric field directs
along the negative X, axis. From Fig. 3, it is seen
that toughness
variation
is symmetric
about
4 = 90”. As the initial poling direction deviates
from 90 degrees, the toughness increases for both
positive and negative electric fields and reaches the
maximum at 0” and 180”. This is in agreement
with experimental observation of anisotropic fracture toughness [2]. For the same initial poling
direction, toughening effect of the positive electric
field is slightly higher near 4 = 90”. As the initial
poling direction deviates from 90 degrees, the
apparent toughness will increase more under the
negative electric field and finally exceed the
toughness under the positive electric field. The two
curves meet at 0” and 180”.
4. TOUGHNESS
VARIATION
FOR
FERROELECTRICS
t
’
20
’ .
40
’ .
60
’ . ’
’
60
109
120
’ . ’
140
160
’
180
4
Fig. 3. Effect of positive and negative electric field on the
toughness
variation
of mono-domain
ferroelectrics
for
different initial poling direction 4.
POLED
To estimate toughness variation for poly-domain
ferroelectrics, a Reuss type approximation, see Hill
[18], is used. Consider a continuum element with
poly-domain structure. The orientation distribution
function of the domains in that continuum element
is denoted by f(q5). The Reuss approximation
assumes that all domains in the continuum element
are subjected to the same applied stress and
electric fields. Their fates of switching are determined by the switching criterion (8) which relates
to their individual orientations. As in all Reusstype assembly, different switching strains are
accommodated
without appreciable local interaction. The collective toughening effect is composed of an integration
over all orientations
(namely with respect to 4) for a specific continuum element, and followed by an area integral for
continuum elements in the entire switching zone.
For an initially homogeneous ferroelectric aggregate, the domain orientation distributions are the
same for all continuum elements. Thus one can
switch the order of integrations. The area integral
will be carried out first and give rise to the
mono-domain
solution discussed in the previous
section. Further integration with respect to all
domain orientations leads to the following superposition integral
AK=
0.51
.
0
FULLY
’ AK($M4)
s -77
d4.
(26)
For unpoled ferroelectric ceramics made up of many
randomly oriented domains, f(4) is uniform for all
the directions. Due to the fact that polar axis of a
domain can only rotate 90” or 180” from its original
direction, it is impossible to align all the domains
along the poling direction. For simplicity, we letf(4)
ZHU and YANG:
be uniform for
[
TOUGHNESS OF FERROELECTRICS
(a) 20
-;,+;
1
4701
r
around the poling direction and vanish elsewhere
a<4<-72
3
4
otherwise
(27)
Substituting mono-domain solution and the orientation distribution function of domains into (26), one
obtains the shielding ratio for fully poled ferroelectries.
K
app=
KU,
(28)
(b)
2.00
r
1 75
t
To evaluate the shielding ratio for fully poled
ferroelectrics, we take the iterative scheme described
in the previous section. Material parameters are listed
in Table 1 and spontaneous strain is assumed to be
0.005. To maintain
the small scale switching
situation, we take the applied electric field from
-0.5E, to +0.5E,. It is instructive to analyze two
extreme cases.
Case 1, Conducting Jaw. In this case, K -+ co, and
the electric field is uniform. As studied by Yang and
Zhu [7], toughness variation will be gauged by the
uniform electric field. In Fig. 4(a) it is seen that
shielding ratio decreases monotonically for positive
electric field, and increases monotonically
for a
negative one. A positive electric field promotes crack
propagation, while a negative one retards it.
Case 2. Impermeablejaw. In this case, K = 0. Hence,
a($) = 1 and the singular electric field dominates.
Figure 4(b) depicts the variation of shielding ratio
with respect to the dimensionless electric field
intensity factor. Under a small electric field, positive
or negative, the shielding ratio increases slightly and
then reaches a maximum. Further increase of the
electric field strength will reduce drastically the
shielding ratio, or even embrittle the material. Except
the slight raise of the shielding ratio at small applied
field, the main effect of the singular electric field is to
degrade the fracture toughness of ferroelectric
ceramics. This prediction agrees qualitatively with the
previous analyses based on the singular electric field,
see Yang and Suo [6], and Lynch et al. [3].
However, no crack is truly impermeable to electric
field, filled as it probably will be by atmosphere or
ionized gases. The actual situation of crack tip
shielding involves both the singular and the uniform
electric fields. As revealed by (28) dimensionless
parameter (K&‘,)/(K,,,y,) measures the relative contribution from the singular electric field, and E/E,
Fig. 4. Toughness variation of fully poled ferroelectrics
under (a) uniform electric field; (b) singular electric field.
(hidden in the expression of function a) the relative
contribution from the uniform electric field. Their
effects are controlled by the relative permeability
parameter K, which is the dimensionless group of
geometric and dielectric constants. Figure 5 demonstrates curves of toughness variation vs applied
electric field under different values of K, where we
take a crack length of 2a = 2 mm and consequently
(K,P,)/(K,,y,)
= 0.57 as E = 0.5E,. The trend of
toughness variation under combined electrical and
mechanical loadings is dictated by the relative
permeability parameter K. With decreasing K, the
effect of the uniform electric field reduces, and the
effect of the singular electric field strengthens. As
shown in Fig. 5, the extreme case of a conducting
crack (in which the toughness variation is controlled
by the uniform electric field) is recovered for the case
of K = 10’. With decreasing K, the effect of singular
electric field becomes appreciable. The extreme case
4702
ZHU and YANG:
TOUGHNESS OF FERROELECTRICS
1.25 -
1.00 -
0.75 -
0.501
46
’
’
-0.4
’
.
-0.2
’
0.0
E/E
’
02
.
’
0.4
*
1
0.6
c
Fig. 5. Shielding ratio as a function of applied electric field
under different values of K.
of an impermeable crack (in which the toughness
variation is dictated by the singular electric field) is
recovered for the case of IC= lo-‘.
Conflicting experimental results about the electric
field effects on toughness variation can be explained
by the above analyses. The geometry of the crack-like
flaw and the permittivity of the material inside the
crack may significantly change the effect of electric
field on toughness. The experimental
data of
toughness variation under electric field by Park and
Sun [lo] gives an asymmetric correlation, suggesting
that the uniform electric field might play a dominant
role near the switching zone boundary. On the other
hand, the effect of the singular electric field might
present in the experiment of Singh and Wang [12].
5. CONCLUDING REMARKS
The present analysis reveals the relevance of
polarization
switch to the variation of fracture
toughness. The effect of switch toughening can be
estimated by (28) for fully poled ferroelectrics. The
analysis indicates that large singular electric field,
positive or negative, usually degrades the fracture
toughness of the materials. While the near tip
uniform electric field has an asymmetric and
monotonic
effect on the variation
of fracture
toughness, toughening under the negative electric
field and embrittling under the positive one. With
moderate complication in the numerical procedure,
this approach can be generated to the cases of
ferroelectrics under arbitrary poling.
The key dimensionless material parameters for the
toughness modeling are revealed. They are the
relative permeability parameter K defined in (3), the
relative strength of the singular electric field b defined
in (14), and the ratio between the elastic strain energy
and the polarization switch energy q, defined in (21).
The effect of the electric field on the fracture
toughness, as indicated in Fig. 5, is rather subtle, and
sensitive to the modeling of flaw boundary condition.
The change of the relative permeability parameter K
may alter the trend of toughness variation with the
applied electric field. Accordingly, a regulation of the
electric boundary condition for the fracture toughness testing is important to get comparable toughness
data.
Acknowledgements-The
authors offer thanks for the
support to this research project by the National Natural
Foundation of China.
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